The function space on an interval is larger than the
interval.

Let F be the set of all functions defined on the
interval (0,1). Such a function associates a
unique number with every number 0 < x < 1. We
do not assume that it has any special properties like being
continuous. (It's a good exercise to show that the
set of continuous functions on (0,1) is in fact
equivalent to the interval (0,1) ) That set is
larger than the set of real numbers in (0,1).

To see this we assume the statement is true and derive a
contradiction. So suppose there is a map

r fr (*)

that associates a function fr wit a number 0 < r <
1, such that for every function f there is a
unique r in (0,1) such that f=fr.
Now construct a function g which is such that
g(r) does not equal fr(r), for all 0 <
r < 1. Such a function can be obtained easily, for
example via the definition

g(r):=fr(r)+1.

Then clearly there is no real number r that is
associated with g and we have a contradiction. An
association (*) does not exist.